Send messages (SMS) or even sending a "tweet" out from Arduino is easy and a few examples of this:http://mydooropener.com/
Monitoring twitter for a very specific message to say turn on a light with an Arduino is still on the simpler side.
(I had a bookmark and code bit that would look for a specific tweet to occur and then do something but not finding that currently but will keep looking). (found it: http://www.instructables.com/id/Twitter-controlled-Arduino-Outputs-no-PCLCD-Disp/
) but not sure if it is still valid see the Arduino support for twitter see: http://arduino.cc/en/Tutorial/TwitterClient
, but as stated that has changed.
But further reading and looking at Temboo will lead to some help and working items.
Also monitoring several accounts Twitter, Facebook as same time gets more complicated especially for random incoming messages specifically to you.
Also look in to GROWL if your an OSX user.
Lastly a Arduino YUN would simplify this with it dual processors as the LINUX side would support the monitoring of accounts with much greater ability and then passing controls to the 32U4. I just bought one of these for ability of greater internet control.